All of the 5th grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$4.50$ each for students, and the group paid $$29.00$ in total. The next month, the same group visited a science museum where the tickets cost $$22.00$ each for teachers and $$8.00$ each for students, and the group paid $$76.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+4.5y = 29}$ ${22x+8y = 76}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-22x-18y = -116}$ ${22x+8y = 76}$ Add the top and bottom equations together. $ -10y = -40 $ $ y = \dfrac{-40}{-10}$ ${y = 4}$ Now that you know ${y = 4}$ , plug it back into $ {5.5x+4.5y = 29}$ to find $x$ ${5.5x + 4.5}{(4)}{= 29}$ $5.5x+18 = 29$ $5.5x = 11$ $x = \dfrac{11}{5.5}$ ${x = 2}$ You can also plug ${y = 4}$ into $ {22x+8y = 76}$ and get the same answer for $x$ ${22x + 8}{(4)}{= 76}$ ${x = 2}$ There were $2$ teachers and $4$ students on the field trips.